[Math Lair] A Logical Paradox

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The following, which illustrates the shopkeeper's paradox, was presented by Lewis Carroll in the July 1894 issue of Mind.

A Logical Paradox

What, nothing to do? said Uncle Jim. Then come along with me down to Allen’s. And you can just take a turn while I get myself shaved.

All right, said Uncle Joe. And the Cub had better come too, I suppose?

The Cub was me, as the reader will perhaps have guessed for himself. I’m turned fifteen—more than three months ago; but there’s no sort of use in mentioning that to Uncle Joe; he’d only say go to your cubbicle, little boy! or Then I suppose you can do cubbic equations? or some equally vile pun. He asked me yesterday to give him an instance of a Proposition in A. And I said All uncles make vile puns. And I don’t think he liked it. However, that’s neither here nor there. I was glad enough to go. I do love hearing those uncles of mine chop logic, as they call it; and they’re desperate hands at it, I can tell you!

That is not a logical inference from my remark, said Uncle Jim.

Never said it was, said Uncle Joe: it’s a Reductio ad Absurdum.

An Illicit Process of the Minor! chuckled Uncle Jim.

That’s the sort of way they always go on, whenever I’m with them. As if there was any fun in calling me a Minor!

After a bit, Uncle Jim began again, just as we came in sight of the barber’s. I only hope Carr will be at home, he said. Brown’s so clumsy. And Allen’s hand has been shaky ever since he had that fever.

Carr’s certain to be in, said Uncle Joe.

I’ll bet you sixpence he isn’t! said I.

Keep your bets for your betters, said Uncle Joe. I mean—he hurried on, seeing by the grin on my face what a slip he’d made—I mean that I can prove it, logically. It isn’t a matter of chance.

Prove it logically! sneered Uncle Jim. Fire away, then! I defy you to do it!

For the sake of argument, Uncle Joe began, let us assume Carr to be out. And let us see what that assumption would lead to. I’m going to do this by Reductio ad Absurdum.

Of course you are! growled Uncle Jim. Never knew any argument of yours that didn’t end in some absurdity or other!

Unprovoked by your unmanly taunts, said Uncle Joe in a lofty tone, I proceed. Carr being out, you will grant that, if Allen is also out, Brown must be at home?

What’s the good of his being at home? said Uncle Jim. I don’t want Brown to shave me! He’s too clumsy.

Patience is one of those inestimable qualities—— Uncle Joe was beginning; but Uncle Jim cut him off short.

Argue! he said. Don’t moralise!

Well, but do you grant it? Uncle Joe persisted. Do you grant me that, if Carr is out, it follows that if Allen is out Brown must be in?

Of course he must, said Uncle Jim; or there’d be nobody in the shop.

We see, then, that the absence of Carr brings into play a certain Hypothetical, whose protasis is Allen is out, and whose apodosis is Brown is in. And we see that, so long as Carr remains out, this Hypothetical remains in force?

Well, suppose it does. What then? said Uncle Jim.

You will also grant me that the truth of a Hypothetical—I mean its validity as a logical sequence—does not in the least depend on its protasis being actually true, nor even on its being possible. The Hypothetical, If you were to run from here to London in five minutes you would surprise people, remains true as a sequence, whether you can do it or not.

I ca’n’t do it, said Uncle Jim.

We have now to consider another Hypothetical. What was that you told me yesterday about Allen?

I told you, said Uncle Jim, that ever since he had that fever he’s been so nervous about going out alone, he always takes Brown with him.

Just so, said Uncle Joe. Then the Hypothetical, if Allen is out Brown is out is always in force, isn’t it?

I suppose so, said Uncle Jim. (He seemed to be getting a little nervous, himself, now.)

Then if Carr is out, we have two Hypotheticals, if Allen is out Brown is in and If Allen is out Brown is out, in force at once. And two incompatible Hypotheticals, mark you! They ca’n’t possibly be true together!

Ca’n’t they? said Uncle Jim.

How can they? said Uncle Joe. How can one and the same protasis prove two contradictory apodoses? You grant that the two apodoses, Brown is in and Brown is out, are contradictory, I suppose?

Yes, I grant that, said Uncle Jim.

Then I may sum up, said Uncle Joe. If Carr is out, these two Hypotheticals are true together. And we know that they cannot be true together. Which is absurd. Therefore Carr cannot be out. There’s a nice Reductio ad Absurdum for you!

Uncle Jim looked thoroughly puzzled: but after a bit he plucked up courage, and began again. I don’t feel at all clear about that incompatibility. Why shouldn’t those two Hypotheticals be true together? It seems clear to me that would simply prove Allen is in. Of course it’s clear that the apodoses of those two Hypotheticals are incompatible—Brown is in and Brown is out. But why shouldn’t we put it like this? If Allen is out Brown is out. If Carr and Allen are both out, Brown is in. Which is absurd. Therefore Carr and Allen ca’n’t be both of them out. But, so long as Allen is in, I don’t see what’s to hinder Carr from going out.

My dear, but most illogical, brother! said Uncle Joe. (Whenever Uncle Joe begins to dear you, you may make pretty sure he’s got you in a cleft stick!) Don’t you see that you are wrongly dividing the protasis and the apodosis of the Hypothetical? Its protasis is simply Carr is out; and its apodosis is a sort of sub-Hypothetical, If Allen is out, Brown is in. And a most absurd apodosis it is, being hopelessly incompatible with that other Hypothetical that we know is always true, If Allen is out, Brown is out. And it’s simply the assumption Carr is out that has caused this absurdity. So there’s only one possible conclusion. Carr is in!

How long this argument might have lasted, I haven’t the least idea. I believe either of them could argue for six hours at a stretch. But, just at this moment, we arrived at the barber’s shop; and, on going inside, we found——


The paradox, of which the forgoing paper is an ornamental presentation, is, I have reason to believe, a very real difficulty in the Theory of Hypotheticals. The disputed point has been for some time under discussion by several practised logicians, to whom I have submitted it; and the various and conflicting opinions, which my correspondence with them has elicited, convince me that the subject needs further consideration, in order that logical teachers and writers may come to some agreement as to what Hypotheticals are, and how they ought to be treated.

The original dispute, which arose, more than a year ago, between two students of Logic, may be symbolically represented as follows:—

There are two Propositions, A and B.

It is given that

  1. (1) If C is true, then, if A is true, B is not true;
  2. (2) If A is true, B is true.

The question is, can C be true?

The reader will see that if, in these two Propositions, we replace the letters A, B, C by the names Allen, Brown, Carr, and the words true and not true by the words out and in we get

  1. (1) If Carr is out, then, if Allen is out, Brown is in;
  2. (2) If Allen is out, Brown is out.

These are the very two Propositions on which Uncle Joe builds his argument.

Several very interesting questions suggest themselves in connexion with this point, such as

Can a Hypothetical, whose protasis is false, be regarded as legitimate?

Are two Hypotheticals, of the forms If A then B and If A then not-B, compatible?

What difference in meaning, if any, exists between the following Propositions?

  1. (1) A, B, C, cannot be all true at once;
  2. (2) If C and A are true, B is not true;
  3. (3) If C is true, then, if A is true, B is not true;
  4. (4) If A is true, then, if C is true, B is not true.

The following concrete form of the paradox has just been sent me, and may perhaps, as embodying necessary truth, throw fresh light on the question.

Let there be three lines, KL, LM, MN, forming, at L and M, equal acute angles on the same side of LM.

Let A mean The points K and N coincide, so that the three lines form a triangle.

Let B mean The triangle has equal base-angles.

Let C mean The lines KL and MN are unequal.

Then we have

  1. (1) If C is true, then, if A is true, B is not true.
  2. (2) If A is true, B is true.

The second of these Propositions needs no proof; and the first is proved in Euc., i, 6, though of course it may be questioned whether it fairly represents Euclid’s meaning.

I greatly hope that some of the readers of Mind who take an interest in logic will assist in clearing up these curious difficulties.